#18061: Implement (correct) action of Atkin-Lehner operators on newforms
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Reporter: pbruin | Owner:
Type: defect | Status: needs_work
Priority: major | Milestone: sage-6.6
Component: modular forms | Resolution:
Keywords: newform Atkin- | Merged in:
Lehner operator | Reviewers:
Authors: Peter Bruin | Work issues:
Report Upstream: N/A | Commit:
Branch: | 48e5d2d924072cdbbaf4051c549e268e8e39e589
u/pbruin/18061-atkin_lehner_action | Stopgaps:
Dependencies: #18068, #18072, |
#18086 |
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Comment (by pbruin):
Hi David,
Thanks a lot for taking a look at this!
> - I think it's not good that the new implementation returns
{{{NotImplementedError}}} on some cases where the ''old'' implementation
was correct and valid:
Fixed by first checking if the Atkin-Lehner matrix is a scalar.
> - The error message returned if Q is not an exact divisor of N is very
uninformative:
> [...]
> I would actually favour handing the special case where Q is a prime
separately (interpreting it as W_p^e^ where p^e^ is the power of p
dividing N). The {{{atkin_lehner_operator}}} method of modular symbol
spaces does something along these lines.
The argument `d` is now handled in the same way as in the
`atkin_lehner_operator` method of modular symbol spaces.
> - This error message is misleading:
> {{{
> ...
> NotImplementedError: action of W_Q is not implemented if a_Q(f) = 0
> }}}
> This action '''is''' implemented in some cases -- when the base ring is
QQ (and it should be implemented in more). We don't want to sell our
capabilities short here.
I changed `if a_Q(f) = 0` to `for general newforms with a_Q(f) = 0`.
> - Your implementation will always fail for odd weights:
> [...]
> Are you trying to renormalise the Atkin--Lehner operator so as to be an
involution in all weights? This can't be done Galois-equivariantly, and is
inconsistent with normalisations elsewhere in Sage. (There's a great quote
from Deligne re these normalisations: "Langlands is very sure he knows
what the square root of p is. I have never been so sure".) I personally
favour the normalisation where W_Q^2^ is Q^(k/2 - 1)^ up to a root of
unity, which '''is''' Galois equivariant.
I was somehow convinced that the normalisation I used was consistent with
the rest of sage, but apparently it isn't. I changed it as you suggested
(I think; did you mean W_Q^2^ = Q^(k - 2)^ up to a root of unity?).
--
Ticket URL: <http://trac.sagemath.org/ticket/18061#comment:14>
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